Q.A displacement vector of magnitude 10 m makes an angle of $30^\circ$ with the positive x-axis. Find its rectangular components $A_x$ and $A_y$.

Just as several vectors can be combined into one resultant, a single vector can conversely be split into components along chosen directions: $\vec V = V_1\hat n_1+V_2\hat n_2+V_3\hat n_3$, where $V_1, V_2, V_3$ are the components along the unit directions $\hat n_1,\hat n_2,\hat n_3$. This splitting process is called resolution. When the chosen directions are mutually perpendicular, the components are called rectangular components — by far the most useful and common choice.

For a vector $\vec R$ in a two-dimensional rectangular coordinate system making angle $\theta$ with the x-axis, the rectangular components are $R_x=R\cos\theta$ and $R_y=R\sin\theta$, so that $\vec R = R_x\hat i+R_y\hat j$. From these, the magnitude is recovered as $R=\sqrt{R_x^2+R_y^2}$ and the direction as $\theta=\tan^{-1}(R_y/R_x)$; the same idea extends naturally to three dimensions with a third component $R_z$ along $\hat k$, giving $R=\sqrt{R_x^2+R_y^2+R_z^2}$.

An important consequence of resolution is the component test for vector equality: two vectors are equal if and only if ALL their corresponding rectangular components are equal ($A_x=B_x$, $A_y=B_y$, $A_z=B_z$). The same logic extends to testing whether two vectors are PARALLEL: they are parallel exactly when one vector's components are a constant multiple of the other's, i.e. when $A_x/B_x=A_y/B_y=A_z/B_z$ for some fixed scalar ratio.